are breaking waves, bores, surges and jumps the same...
TRANSCRIPT
ORIGINAL ARTICLE
Are breaking waves, bores, surges and jumps the sameflow?
Pierre Lubin1,2 • Hubert Chanson2
Received: 26 July 2015 / Accepted: 21 July 2016 / Published online: 1 August 2016� Springer Science+Business Media Dordrecht 2016
Abstract The flow structure in the aerated region of the roller generated by breaking
waves remains a great challenge to study, with large quantities of entrained air and tur-
bulence interactions making it very difficult to investigate in details. A number of
analogies were proposed between breaking waves in deep or shallow water, tidal bores and
hydraulic jumps. Many numerical models used to simulate waves in the surf zone do not
implicitly simulate the breaking process of the waves, but are required to parameterise the
wave-breaking effects, thus relying on experimental data. Analogies are also assumed to
quantify the roller dynamics and the energy dissipation. The scope of this paper is to
review the different analogies proposed in the literature and to discuss current practices. A
thorough survey is offered and a discussion is developed an aimed at improving the use of
possible breaking proxies. The most recent data are revisited and scrutinised for the use of
most advanced numerical models to educe the surf zone hydrodynamics. In particular, the
roller dynamics and geometrical characteristics are discussed. An open discussion is
proposed to explore the actual practices and propose perspectives based on the most
appropriate analogy, namely the tidal bore.
Keywords Breaking waves � Breaking bores � Hydraulic jumps � Air bubble entrainment �Flow singularity � Tidal bores
1 Introduction
Surface wave breaking, occurring in the open ocean or the coastal zone, is a complex and
challenging two-phase flow phenomenon which plays an important role in numerous
processes, including air–sea transfer of gas, momentum and energy, and in a number of
& Pierre [email protected]
1 Universite de Bordeaux, I2M, CNRS UMR 5295, 16 Avenue Pey-Berland, 33607 Pessac, France
2 School of Civil Engineering, The University of Queensland, Brisbane, QLD 4072, Australia
123
Environ Fluid Mech (2017) 17:47–77DOI 10.1007/s10652-016-9475-y
technical applications such as acoustic underwater communications and optical properties
of the water column. The major visible feature during wave breaking is the large quantities
of air entrained in the form of bubble clouds and whitecaps, generally coined surface foam
(Figs. 1, 2). The generation of bubble clouds has been shown to induce energy dissipation
and turbulent mixing, to contribute to heat exchange and enhance gas transfer [94, 182].
Bubble clouds have been shown to influence climate and intensification of tropical
cyclones [176], and cause the ocean ambient noise [148]. The breakup and evolution of
entrained air into numerous bubbles is a source of acoustic noise, which is important for
naval hydrodynamics. The hydrodynamic performance of ships is influenced by the wake
Fig. 1 Sketches of breaking waves, bore, roller and hydraulic jump
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modified by the air entrainment, and the sound generated by the bubble clouds render the
ships subject to detection. In hydraulic engineering, large spillways are often protected
from cavitation damage by controlling aeration [76, 159].
(A)
(B)
(A1)
(A2)
(B1) (B2)
(C)
Fig. 2 Air entrainment in breaking waves, bores and jumps. a Breaking waves. a1 Plunging breaking offMain Beach, North Stradbroke Island (Australia) on 19 April 2014—Shutter speed: 1/400 s, a2 spillingbreaker off Pointe du Grand Minou (Brest, France) on 18 March 2004—Shutter speed: 1/800 s. b Breakingtidal bore of the Qiantang River (China) on 11 October 2014. b1 Bore approaching the old seawall atXinchang at 12:50, b2 Bore roller propagating (from left to right) parallel to the seawall at Qilimiao at13:20—The roller height was about 2.5 to 3 m, the bore celerity 4 to 5 m/s and Fr *2—Shutter speed:1/6400 s. c Hydraulic jump roller (steady flow direction from right to left): Fr = 7.5, Re = 1.4 9 105,d1 = 0.03 m, shutter speed: 1/1000 s
Environ Fluid Mech (2017) 17:47–77 49
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Many numerical models (e.g. Boussinesq equations) used to study waves in the surf
zone do not implicitly simulate the breaking process of the waves [54]. The wave-breaking
effects have to be parameterised by incorporating additional terms in the mass and
momentum equations (e.g. [13, 55, 100, 141, 171]. The challenge is to take the breaking
process into account to ensure an accurate description of the surf zone, including the wave
height decay and the setup development. The main consideration is to dissipate energy
when wave breaking is likely to occur. Svendsen [167, 168] proposed the roller concept, in
the form of a volume of water carried shoreward with the wave. Local roller thickness and
mean front slope of the breaker were used to quantify part of the local momentum deficit.
But the vertical surface roller of the breaking wave is only considered to play an important
part in the momentum and energy conservation. However, the energy flux and dissipation
during wave breaking remain difficult to quantify. Most recent modelling attempts are still
struggling with the lack of physical knowledge of the finest details of the breaking pro-
cesses, which makes the task of parameterising breaking effects very difficult since no
universal scaling laws for physical variables have been proposed so far. Physical param-
eters, such as the height and length of the roller, have to be quantified and criteria have to
be defined with critical bounded values to estimate where the waves break and stop
breaking. Thus models still need calibration and further improvements [22].
The turbulent flow dynamics in bubble clouds is a very challenging numerical problem.
Esmaeeli and Tryggvason [75] studied direct numerical simulations of buoyant bubbles in
a two-dimensional periodic domain. They simulated 144 and 324 bubbles, showing that the
work done by the buoyant bubbles increased the energy of flow structures much larger than
the bubbles. But 3D direct modeling of air bubble entrainment and evolution at the scale of
the surf zone is computationally unaffordable. Another way of tackling dispersed two-
phase flows is using a continuum-mechanical approach [71]. Two-fluid models are used to
model the polydisperse two-fluid bubbly flow based on mixture theory
[26, 68, 127, 134, 162]. A first attempt to use a continuum type model for studying bubbly
flow under surface breaking waves was made by Shi et al. [162]. They proposed a phys-
ically-based numerical model for prediction of air bubble population in a surf zone-scale
domain. The air entrainment was formulated by connecting the shear production at air–
water interface and the bubble number density with the bubble size spectra as observed by
Deane and Stokes [67]. The model was initially fed with the entrained bubbles and used to
simulate the evolution of the bubble plumes. This approach requires much less spatial and
temporal resolution than needed to capture detailed air entrainment process in DNS sim-
ulations. The model results revealed that bubbles larger than 1 mm provide a major
contribution to void fraction, while smaller bubbles contribute significantly to the cumu-
lative interfacial area of the bubble cloud but do not contribute much to the total volume of
air. Discrepancies between observations and model behaviour were nevertheless reported.
Based on the works of Ma et al. [127], Derakhti and Kirby [68] used an Eulerian–Eulerian
polydisperse two-fluid model in an LES framework. Detailed overviews on methods and
models for CFD of multiphase flows can be found in textbooks [61, 72]. More information
about turbulence modelling in the framework of multiphase flows is given by Labourasse
et al. [107] and Bombardelli [17]. Smoothed-particle hydrodynamics (SPH) is also a mesh-
free method which can be used to describe accurately the 3D surf zone hydrodynamics, as
recently shown by Farahani and Dalrymple [77] who investigated some novel coherent
turbulent vortical structures under broken solitary waves. The state-of-the-art is detailed by
Gomez-Gesteira et al. [84] and Violeau and Rogers [177], who detailed a number of
examples in which SPH simulations have been successfully used in fluid flow research and
hydraulic engineering.
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Numerical models still rely on experimental data. Detailed information on the
temporal and spatial variations of the void fraction fields beneath breaking waves is
required. Instantaneous void fraction and interfacial velocity data are critically needed
to calibrate and improve numerical models of the two-phase flow generated beneath
plunging and spilling breaking waves. Models for air-entrainment are critically
dependent upon accurate estimates of the surface area affected by wave breaking.
Controlled laboratory experiments and accurate measurements of void fraction and
bubble size distributions beneath plunging and spilling breakers are still very chal-
lenging. When a wave breaks, the tip forms a liquid jet which impinges on the front
face of the wave and creates an air cavity which breaks into bubbles. The character-
isation of the bubble sizes resulting from the cavity collapse has to be measured and
the trajectories of these entrained bubbles are also critical information. The initial
stages of the breaking of a wave generated a large amount of bubbles production and
to the distribution at greater depths. The bubble clouds will then form, grow and decay
during the propagation of the turbulent air/water mixing region forming the bore, the
temporal variations of all bubble cloud dimensions reflecting this evolution. The large
volumes of air in bubbles rapidly evolve into a distribution of bubble sizes which
interacts with liquid turbulence and organised motions.
The motion of bubbles relative to the liquid causes velocity fluctuations in the
water column and increases the energy of liquid motion at the scales comparable with
the bubble diameter [68]. Bubble plume kinematics and dynamics, and the structure of
the turbulent bubbly flow under breaking waves constitute critical information to be
taken into account for an accurate description of the wave breaking process [129].
While the former can be studied experimentally, the liquid–bubble interactions, i.e. the
effects of dispersed bubbles on organised and turbulent motions, are still poorly
understood.
When looking at a bore, whereas it has been generated by a stationary hydraulic jump, a
surface wave breaking on the ocean or in the surf zone, or a tidal bore propagating
upstream a river, the question is: are we looking at the same flow? Is there only one bore
structure, or are there variations depending on the initial conditions leading to its occur-
rence and behaviour? To what extent can we compare the bores and use the quantities
through similarity? It is the aim of this contribution to contribute to the transfer of
knowledge from detailed measurements realised in hydraulic jumps and tidal bores, to the
wave breaking investigation. The first part of the article is dedicated to the identification of
the knowledge gaps encountered when attempting to simulate numerically the hydrody-
namics of breaking waves and a review of the various analogies which have been proposed
in the literature. The next part reports on the state-of-the-art of the studies focusing on the
void fraction and velocity analysis under breaking waves, tidal bores and hydraulic jumps.
Based on this survey, we attempt to identify and assess the quantities which can be
considered for possible analogies. The most recent data are revisited and scrutinised for the
use of most advanced numerical models to educe the surf zone hydrodynamics, highly
influenced by the wave breaking process. An open discussion is proposed to explore the
actual practises and propose perspectives based on the most appropriate analogy, namely
the tidal bore.
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2 Knowldege gaps for the modelling of the surf zone hydrodynamics
2.1 Current state of practice in numerical modelling and limitations
Most numerical models only consider macro-scale roller properties. The roller formation
and propagation have been shown to be a highly unsteady process, with air entrainment and
turbulence generation. The most advanced models, which are generally used to simulate
non-linear wave transformations in coastal areas, are based either on the non-linear shallow
water equations (NSW), the Boussinesq-type equations (BT), or some form of hybrid
model. Extensive developments and break-through progress have been made recently for a
large variety of coastal engineering applications (e.g. [3, 22, 100, 171]). A key feature, the
breaking process, is however not explicitly simulated and missing in these models. Several
approaches and parametrisations have thus been proposed to introduce wave breaking in
NSW and BT models. Any such approach requires the quantification of energy dissipation,
dynamically activated when wave breaking is likely to occur. Some physically based
criteria have to be able to activate or deactivate these extra terms and simple expressions
are generally favoured. Simple quantities include geometrical aspects of the roller,
including heights, lengths and angles, easily extracted from any visual observations in
laboratory and in the field, All these quantities cannot be estimated from a single exper-
iment. Instead a composite set of data and practices have been elaborated though time by
looking at various analogue flows, and some variations have been proposed in order to fill
the gaps.
Practically, most numerical models need to evaluate:
1. A Froude number Fr characteristic of wave breaking, of when it occurs and stops (with
Fr varying with water depth). Currently, an accepted value for the transition between
non breaking and breaking waves has been identified in Froude number range between
1.3 and 1.6 [145]. This is based upon the analogy with non-breaking undular hydraulic
jump and bore [46, 78, 104, 117, 173];
2. The roller height hr, derived from momentum considerations (see Appendix 2);
3. The roller length Lr, determined empirically. A common parameterisation is
Lr = 2.91 9 hr [87], although the re-analysis of large-scale experiments suggests
Lr/hr & 1 to 8 (Fig. 5). In Fig. 5, steady breaker, stationary hydraulic jump and tidal
bore data are compared;
4. The mean front slope angle / [161], typically between 8� and 30� for the termination
and initiation of the breaking event respectively;
5. The roller celerity (or celerity of the breaking wave);
6. The energy dissipation in the roller region;
7. The bubble size distributions, often improperly estimated based upon Hinze’s [89]
model developed in the case of a single droplet under non-coalescing conditions (!).
To estimate most of these quantities, flow analogies have been considered, but some
limitations are clearly identified and some modifications, based on new experimental data
analysis, are proposed in the following sections.
2.2 Flow analogies or not?
A number of analogies were proposed between breaking waves, bores and jumps (Ap-
pendix 1). Appendix 1 lists a number of early seminal references and Fig. 1 presents
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definition sketches. The steady breaker configuration was proposed as a simplification of
the spilling breaker [6, 7]. Important results were obtained [8, 59, 63, 73, 121], but there is
still on-going argument about the validity of this analogy [102]. Further links were
developed between breaking waves and steady flow configurations. These encompassed
comparisons between plunging breakers and plunging jets [44, 47, 48, 56, 93, 143, 160],
between spilling breakers and stationary hydraulic jumps [23, 24, 123, 128, 147] and
between spilling breakers and translating hydraulic jumps (also called positive surges or
tidal bores) [24, 123, 147]. In parallel, there have been numerous discussions about the
similarities and differences between stationary and translating hydraulic jumps (e.g.
[65, 118, 164, 174], although the open channel hydraulic literature develops the same
integral approach for both types [32, 41, 88, 118].
To date, the mechanistic connections between these flows are not well understood and
have not always been successful. Wave-plunging jet conditions appear to produce a
qualitatively different type of impact, with almost no penetration into the oncoming flow
and a pronounced splash that cascades multiple times down the face of the wave. What is
better characterised however, is the volume of air trapped by the initial contact of the jet
with the wave face, which has been numerically simulated, and its shape has been suc-
cessfully modelled, at least for a limited set of conditions (e.g. [126]. However, the
processes that follow the initial contact are only known qualitatively for the majority of the
breaking conditions, and thus still require further study in order to acquire improved
physical understanding. Furthermore, wave breaking is a combination of transient pro-
cesses which evolve within the breaking duration making adequate physical understanding
a challenging proposition. Overall, ‘‘In studying any turbulent flow it is very helpful if it
can be shown to be similar to other well known flows’’ [147]. Below, a number of seminal
flow configurations are explored and the relevance of flow comparisons is discussed.
3 Void fraction kinematics
3.1 Breaking waves
Fuhrboter [79] discussed the correlation between the turbulence generated in the surf zone
and the amount of air entrained during the breaking of the waves, as well as the sudden
reduction of wave height and energy. He highlighted the importance to study quantitatively
the air entrainment process for a detailed comprehension of the surf zone physics. Vagle
and Farmer [175] and recently Anguelova and Huq [1] reviewed the different techniques
used to quantify the void fraction under breaking waves. Both works concluded that
combined techniques were the best approach. Indeed, the higher the concentrations of
bubbles within bubble clouds, the more difficult it is to count and measure individual
bubbles.
Some studies have been conducted in field while others have been completed in physical
wave tanks. Thorpe [170] studied wind-waves breaking and speculated that wind speed,
salinity, and temperature were major factors, possibly responsible for existing discrepan-
cies that arised when comparing data from different sources. Monahan [133] proposed the
terms Alpha-plume (high void fraction, short lifespan), Beta-plume, and Gamma-plume
(low void fraction, long lifespan) to describe the evolution of a bubble cloud, from its
formation to its disappearance (e.g. dissolution, degassing and advection). Most field
studies confirmed that the Alpha-plumes consist of high void fractions (10 % or more) with
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large bubble sizes (radii ranging from tens of micrometers to millimeters). At the other end
of the process, the Gamma-plume were observed to be very low void fraction between
10-5 and 10-8 and containing bubbles with radii on the order of O(10–100) lm. The
lifetime of a whole bubble cloud may be about a hundred of seconds. The bubble clouds
are also generally confined to the first few meters of the water column. For example,
Lamarre and Melville [110] compared field and laboratory void fraction measurements
obtained with an impedance probe, and showed large void fraction values at shallow
locations while lower void fraction values were found deeper. Deane [66] used acoustic
and optical measurements of individual breaking waves in the surf zone, off La Jolla
Shores beach, California. Total void fractions of 0.3–0.4 were measured, consisting of
bubbles with radius greater than 1 mm. Stokes and Dean [165] observed that the time scale
for the generation of clouds of submillimetric bubbles was on the order of about 90 ms.
Dahl and Jessup [64] found comparable quantities in deep-ocean studies. Gemmrich and
Farmer [83] measured void fraction values (e.g. 10-2 at 0.25 m below the free-surface),
associated with low penetrating breaking events (spilling breakers), while they speculated
that higher values of void fractions found deeper would be associated with more energetic
violent events (plunging breakers). Interestingly Gemmrich [82] found higher turbulence
levels within the wave crest region of the breaking waves, suggesting that the bubble
fragmentation process is mainly driven by turbulence. Most studies reported that void
fraction changes significantly during the lifetime of the bubble cloud, from high void
fractions in the first seconds of the breaking event to residual void fractions persisting for
long times. Most field studies consisted in wind-waves breaking observations, with only
few events giving data susceptible to be accurately analysed.
A lot of studies investigated the hydrodynamics in the surf zone, especially the general
mechanisms involved during the breaking process [146], the generation of turbulence [10],
and sediment transport. When waves break, the flow suddenly exhibits a violent transition
from irrotational to rotational motion over the entire water column. Two main types of
breaker types have been studied: (1) the spilling breakers, where white foam, consisting of
a turbulent air/water mixture, appears at the wave crest and spills down the front face of the
propagating wave; and (2) the plunging breakers, where the front face of the steepening
wave overturns and impacts the forward face of the wave. These two breaker types have
been shown to have similar initial motions, but with different length scales [9]. When
approaching a beach, the waves change form due to the decrease in water depth. The
forward face of the wave steepens and the wave becomes asymmetric. Once the front face
becomes almost vertical, a jet of liquid is projected from the crest of the wave. The tongue
of water thrown from the crest develops and free falls down forward into a characteristic
overturning motion, and eventually hits the water at the plunge point. Depending on the
position of the plunge point, different breaker types can be observed. If the plunge point is
located very near to the crest of the wave, the resulting splash is directed down the wave
leading to a spilling breaker. Otherwise, if the jet is ejected farther towards the lower part
of the face of the steepening wave, the wave becomes a plunging breaker. The plunging jet
encloses an air pocket when it finally hits the wave face at the plunge point. The jet re-
enters the water after impact, forcing up a second jet, called splash-up. The early works of
Miller [132], Basco [9], Jansen [97] and Bonmarin [18] were dedicated to qualitative
description of the dynamics of the breaking process, the air entrainment and the evolution
of the large-scale geometric properties of bubble plumes. The overturning process, sub-
sequent overturning motion and plunging jet impact were described, resulting in the
identification and tracking of breaker vortices trajectories. Some information about the
evolution (size, shape and position) of the bubble plumes were also detailed. The jet-splash
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cycles, occurring several times in a single breaker, have been shown to be responsible for
the generation of a sequence of large-scale vortices with a horizontal axis of rotation, some
of these eddies have been shown to be co-rotating vortices and some counter-rotating
vortices depending on the splash-up mechanism [18, 132]. Nadaoka et al. [142] detailed the
flow field under a turbulent bore propagating towards the shoreline, resulting from a
spilling breaking wave. Large-scale horizontal eddies are present in the bore front, while
behind the wave crest the flow structure changes rapidly into obliquely downward stretched
three-dimensional (3D) eddies, so-called ‘obliquely descending eddies’. Lin and Hwung
[120], Govender et al. [85] and Kimmoun and Branger [101] also described the large
motions of aerated regions under plunging breaking waves, with splash-ups and vortical
structures.
Miller [132] measured the average bubble concentration in plunging and spilling
breakers and indicated a larger bubble density presence in plunging breakers (about 31 %
in the late stage compared to 19 % for spilling breakers); these results were in agreement
with earlier descriptions from Miller [131]. Lamarre and Melville [109] concluded that a
large portion of the mechanical energy of the wave was lost in entraining the bubble
clouds. High values of void fractions (up to 100 %) were found next to the free-surface,
and void fractions of at least 20 % were observed for up to half a wave period after the
breaking occurrence. They later confirmed that air entrainment was closely related with the
energy dissipation of the breaking wave [111]. Several other works provided more com-
prehensive laboratory measurements of the void fraction in breaking waves, detailing the
vertical and horizontal distributions of void fraction. Cox and Shin [60] measured the void
fraction in the aerated region at a point using a capacitance probe, and observed peak
ensemble-averaged void fractions in the range of 15–20 %. Surprisingly, they measured
higher void fractions under the spilling breakers than under the plunging breakers. The
temporal variation of void fraction, above and below the still water level, was analysed
using three breaker types (spilling, spilling/plunging and plunging). The temporal variation
of void fraction above the still water normalized by the wave period and average void
fraction appears to be remarkably self-similar (independently of the breaker type). Hwung
et al. [94] found a deeper penetration of air bubbles under plunging breaking waves and
higher void fractions (18 %), compared to the spilling breaking waves (12 %). Similarly
Hoque and Aoki [90], using a conductivity probe, measured maximum void fractions of 20
and 16 % beneath plunging and spilling breakers, respectively. Mori et al. [135] obtained
void fractions of 19 % beneath spilling breaking waves and 24 % beneath plunging
breaking waves, using dual-tip resistivity void probe. Interestingly they also studied scale
effects according to Froude similarity and using two different scaled experiments. Void
fractions were affected by the geometric scale, with larger quantities being found in the
larger experiment, while the bubble size spectra proved to be nearly independent. Kim-
moun and Branger [101] estimated the evolution of void fractions using particle image
velocimetry images and velocity measurements. They reported large void fractions of up to
88 % in the first splash-up location, decreasing slowly when the breaking wave propagates
towards the shore, with values between 20 and 30 %. Much lower void fractions were
found in other studies. Kalvoda et al. [98] investigated the geometric and kinematic
characteristics of large air bubbles clouds produced by spilling breaking wind waves. They
observed that the lifetime of the bubble cloud was about 1.4 times the wave period, with
bubbles diameters in the range of 1.0–10 mm. They found a void fraction of about 0.4 %.
Leifer et al. [113] reported void fractions between 0.2 and 2.3 % beneath breaking wind
waves, using a video system to characterise the bubble clouds. Blenkinsopp and Chaplin
[14] studied plunging, spilling/plunging and spilling breaking waves. They calculated
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integral properties of the bubble clouds and splash-ups, such as areas and volumes of air
entrained, trajectories of centroids and energy dissipation, and showed remarkable simi-
larity between plunging and spilling breakers. Their data indicated that the evolution of the
bubble clouds was subjected to scale effect. Rojas and Loewen [156] detailed the void
fraction evolution in spilling and breaking breakers. They observed that beneath plunging
breaking breakers, the mean void fraction ranged between 1.2 and 37 %, while beneath
spilling breaking waves, the mean void fraction ranged between 17 and 29 %. They found
that ‘‘an energetic spilling breaker may entrain approximately the same volume of air as a
steeper, larger-amplitude plunging breaker’’. They identified and tracked successive
bubble clouds, detailed the void fractions at each step of the breaking events, and found
that, beneath the spilling breaker, the celerity of the bubble cloud compared with the phase
speed. Beneath the plunging breaker, the celerity of the air cavity was about 70 % of the
phase speed. This has to be compared to the celerity of the bubble cloud entrained by the
propagating splash-up which has been measured to be about 90 % of the phase speed.
Blenkinsopp and Chaplin [14] and Rojas and Loewen [156] found that the volume of air
entrained by the splash-up, observed during a plunging breaking event, was greater than
the volume of air entrained by the initial plunging jet (about 60 % more). More recently
Anguelova and Huq [1] used an imaging technique to quantify the phase dependent void
fraction, and measured values reaching 80–99 % at the wave crest phase and decreasing to
20–30 % at the trough phase. Lim et al. [119] confirmed these results in the case of a
plunging breaker. They showed that the distribution of the turbulent intensities matched the
vorticity and void fraction fields. Nevertheless, some differences could be observed in the
experimental results for the peak values of void fraction, indicating a strong temporal and
spatial variability in the unsteady breaking waves [119]. The difference in the locations of
the measurements and the method used to generate the breaking could also be responsible
for the discrepancies. Some authors indicated that the mean void fraction could be mod-
elled by a linear function of time followed by an exponential decay. Hoque and Aoki [90]
found that the void fraction distribution followed the analytical solution of an advection
equation. This is not surprising as there is a general consensus about in void fractions
contours in the breaking waves, shapes and general kinematics of the aerated regions.
However, some differences can also be noted. Lamarre and Melville [109] found that that
the temporal variation of the normalized void fraction in deep water breakers could be
fitted by a power law t-2.3, while Rojas and Loewen [156] found the decay of void fraction
followed a power law t-2.6. Lim et al. [119] reported similar findings. Cox and Shin [60]
found a linear growth followed by exponential decay, but with a different rate, which could
be attributed to the shallow-water conditions used in their works.
3.2 Tidal bores
Undular tidal bores are observed for Froude numbers less than 1.3–1.4, and breaking tidal
bores with a marked roller are seen for Froude numbers large than 1.4–1.6 [37, 104].
Velocity measurements in breaking tidal bores were performed using particle image
velocimetry and acoustic Doppler velocimetry with most data obtained in the clear-water
column below the aerated roller region and for Froude numbers less than 2.5
[38, 69, 91, 105]. The data indicated a sharp deceleration of the flow associated with the
bore passage, as well as large fluctuations of all velocity components during and shortly
after the bore roller passage [105, 114].
Air–water flow measurements in tidal bores are rare. Leng and Chanson [114] high-
lighted some distinctive air bubble entrainment at the toe of the roller. Their data showed a
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substantial number of bubbles with millimetre sizes: i.e., between 1 and 5 mm, with larger
bubbles detected at higher vertical elevations in a more intermittent manner. The same data
set showed that the toe perimeter shape fluctuated rapidly with transverse distance and
time, and large fluctuations in bore celerity were observed. That is, the celerity of the roller
toe fluctuated rapidly with both time and transverse distance, although in a quasi-two-
dimensional manner on average [115]. Yet both laboratory and field observations hinted to
the existence of large-scale three-dimensional bubbly structures [43, 114, 116].
3.3 Hydraulic jumps
A hydraulic jump is the sudden and rapid transition from a supercritical to subcritical flow,
characterised by the development of large-scale turbulence, surface waves and spray,
energy dissipation and air entrainment (Figs. 1, 2). Undular hydraulic jumps, observed for
Froude numbers less than 2–3, are not discussed herein. The breaking jump is a turbulent
shear flow [92, 150, 157], and vorticity is advected in the air–water shear zone. The first
successful air–water flow measurements were conducted by Rajaratnam [149]. Using hot
film anemometry, Resch and Leutheusser [153] and Babb and Aus [2] measured distri-
butions of Reynolds stresses and turbulence intensity in hydraulic jump rollers. High-
quality data were obtained using dual-tip phase-detection probes, allowing significant
improvements on the characterisation of the air–water flow field inside the jump roller.
Most studies were conducted for inflow Froude numbers greater than 5 [39, 49, 178].
Relatively Fewer studies investigated hydraulic jumps with lower Froude numbers
[29, 140]. Other research works also used flow visualization techniques [112, 137].
All experimental investigations in breaking jumps showed some intense air entrainment,
with singular air entrapment at the roller toe and interfacial aeration across the roller free-
surface [178]. Air is entrapped at the discontinuity between the impinging flow and the
roller: i.e., the roller toe (Figs. 1, 2). The roller toe perimeter is a line source of air bubbles,
as well as a line source of vorticity [31, 91]. In the shear region downstream of the roller
toe, the local maximum in void fraction Cmax decays longitudinally in a quasi-exponential
manner [49, 140]. A key feature is the developing shear layer and the recirculation region
above. The turbulent shear flow is somehow analogous to a wall jet although basic dif-
ferences between ideal wall jet and hydraulic jumps include the effect of longitudinal
pressure gradient qP/qx\ 0, conservation of mass, and the natural forcing which is likely
to enhance the two-dimensionality and periodicity of coherent structures [99]. In the jump,
the maximum velocity Vmax decays longitudinally as x-1/2 [39, 49, 139].
Figure 3 presents some typical air–water flow measurements in hydraulic jumps with
breaking roller. Figure 3a shows longitudinal variations in vertical distributions of void
fraction and interfacial velocity in the roller, where C is the time-averaged void fraction, V
is the interfacial velocity, x is the longitudinal distance from the roller toe, Lr is the roller
length, z is the vertical elevation, and d1 and V1 are the inflow depth and velocity
respectively. Overall the measurements highlighted two distinct air–water flow regions: (1)
the air–water shear layer, and (2) the upper free-surface region above. The shear layer is
characterised by strong interactions between the entrained air bubbles and turbulent
coherent structures, associated with a local maximum in void fraction, bubble count rate
and longitudinal velocity. In the shear layer, the distributions of void fractions follow an
analytical solution of the advection–diffusion equation for air bubbles assuming singular
air entrapment at the roller toe:
Environ Fluid Mech (2017) 17:47–77 57
123
(A)
C/10, V/V1, (x-x1)/d1
z/d 1
-0.05 0.05 0.15 0.25 0.35 0.45 0.550
1
2
3
4CV/V1Z90/d1
(B)
C
z/d 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7(x-x1)/d1=15.2(x-x1)/d1=11.4(x-x1)/d1=7.59(x-x1)/d1=3.80
Singular aerationInterfacial aeration
(C)
V/Vmax
(z-Z
Vm
ax)/Z
0.5
-0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.2-0.5
0
0.5
1
1.5
2
2.5
3Wall jetx/Lr=0.085x/Lr=0.169x/Lr=0.254x/Lr=0.394x/Lr=0.508x/Lr=0.677
58 Environ Fluid Mech (2017) 17:47–77
123
C ¼qairq
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4� p� D� x0p � exp �
z0�1ð Þ2x0
4� D
!
þ exp �z0þ1ð Þ2x0
4� D
! !
ð1Þ
where x0 = x/d1, z0 = z/d1, D is a dimensionless diffusion coefficient, q = V1 9 d1, and
qair is the entrained air flow rate per unit width. D and qair/q are derived from best data fit.
In the upper free-surface region above, the void fraction increases monotonically with
increasing distance from the bed from a local minimum up to unity, following an analytical
solution of the advection–diffusion equation for interfacial aeration/de-aeration:
C ¼ 1
2� 1� erf
1
2�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
V1 � x
Dt
r
� z� Z50
x
� �� �
ð2Þ
where Dt is a diffusion coefficient, Z50 is the elevation where C = 0.5 and erf is the Error
function. Both Eqs. (1) and (2) are compared with experimental data in Fig. 3b, with solid
lines and dashed lines respectively. The interfacial velocity distribution presents a self-
similar profile close to that of wall jet:
V� Vrecirc
Vmax � Vrecirc
¼ exp �0:88� z� zVmax
z0:5
� �2 !
zVmax \z ð3Þ
where Vmax is the maximum velocity in the shear region, Vrecirc is the negative recircu-
lation velocity, z0.5 is the elevation where V = (Vmax ? Vrecirc)/2 and zVmax is the ele-
vation where V = Vmax. Equation (3) is compared with experimental data in Fig. 3c. Note
the negative time-averaged recirculation velocity, with Vrecirc/Vmax *-0.4 to -0.6 on
average [39, 185] (Fig. 3c). Further observations include the integral turbulent length scale
Lt characterising the size of large turbulent eddies advecting the bubbles in the hydraulic
jump roller [34, 179]. Typical results highlight the relationship between integral turbulent
length scale and inflow depth, with on average Lt/d1 * 0.5. Moreover, several studies
were dedicated to the analysis of the clustering process in a hydraulic jump [86], focussed
on the interaction between air bubbles and turbulence, and confirming the influence of the
bubble clustering on the surrounding flow field whereby hydrodynamic interactions and
enhanced velocity fluctuations can play an important role.
4 Discussion
Based on the previous section, we propose a discussion on limitations, disagreements and
obstacles which have to be overcome to improve the knowledge of the turbulent air–water
dynamics encountered in breaking waves and bores. We focus on the flow aeration and
bubble sizes distribution in the water column, water salinity and the definition of the
Froude number.
bFig. 3 Air bubble entrainment in a hydraulic jump with a marked roller. a Longitudinal variations ofvertical distributions of void fraction and interfacial velocity—Data set: Chachereau and Chanson [29],d1 = 0.044 m, Fr = 3.1, Re = 8.9 9 104. b Vertical distributions of void fraction—Data set: Chachereauand Chanson [29], d1 = 0.0395 m, Fr = 5.1, Re = 1.3 9 105. c Vertical distributions of interfacial velocityin the roller—Data set: Chanson [37], d1 = 0.018 m, Fr = 9.2, Re = 6.9 9 104
Environ Fluid Mech (2017) 17:47–77 59
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4.1 Flow aeration and bubble sizes
Very few detailed studies on air entrainment induced by breaking waves exist (Sect. 3.1).
A number of issues remain and definitive conclusions have not yet been drawn. Wave
conditions and characteristics are not easily comparable. For example, some studies were
centred on deep-water breaking waves while others focused on depth-limited conditions,
some breaking waves were mechanically generated, others were wind-waves. Further the
metrology (optical, acoustic, laser, conductivity BIV, PIV, video, photographs) and
experimental apparatus (single or double probes, measurements locations, assumptions for
analysis, mixture density estimations for wave characteristics, etc.) often suffered from
limitations and inaccuracies (e.g. [27, 33, 60]). Phase-detection needle probes are intrusive
point measurements, while BIV or PIV are non-intrusive but remain inaccurate when
operated in highly aerated turbulent flows (bubble clustering, shape changes, light
reflection, image processing technique and analysis). Sampling rate is another issue when
trying to characterise turbulent bubbly flow properties. A number of studies thus used
combined techniques to ensure a certain level of confidence in their measurements.
Several findings may be highlighted. All authors reported that the energy of the waves
was dissipated during the early stages of the breaking events, due to the splash-up kine-
matics, turbulence generation and air entrainment process. The kinetic energy was found to
decay as t-1, both experimentally [70, 109, 110, 130, 152] and numerically
[52, 95, 96, 108, 125, 126]. The same decay rate was observed for all breaker types
(spilling, weak plunging and plunging) [126, 130], while similar integral properties in
terms void fraction were comparable for different types of breaking waves [14, 60].
Locations of air entrapment in the roller can be identified at the toe of the advancing roller,
and on the whole interface of the roller, where multiple splashes are observed
[18, 119, 120, 132, 142, 156]. Large coherent eddies play a significant role in turbulent
production due to the large velocity gradients and contribute to shear in the mean flow and
fluctuations.
The highest velocities and turbulence intensities are found in the front part of the wave
crest, at the toe of the wave roller and in the shear regions. In both spilling and plunging
waves, the propagation of the roller to the shore line is associated with higher levels of
turbulence [85]. The breaking bore is widely recognized as being the primary source of
turbulence under breaking waves. The void fraction appeared to be correlated with the flow
turbulence and turbulent kinetic energy [60, 135], but experimental results have not yet
proved that the turbulent properties are enhanced by high concentrations of bubbles, and
vice and versa. Indeed, the relations between void fraction and bubble size distributions,
and between void fraction and turbulence, remain unclear: ‘‘the entrained bulk of air and
bubbles generate turbulence which breaks coarser bubbles into finer ones in this phase’’
[136]. Gemmrich [82] speculated that the bubble fragmentation process is mainly driven by
turbulence. But the bubble dynamics might also be a major source of turbulence in the
surrounding fluid.
Altogether breaking waves can produce large millimetre bubbles, although smaller
bubbles may be found in the breaking region (Table 1). Table 1 summarises a few detailed
observations. Larger bubbles are mainly present in the roller region, while much smaller
bubbles with large residence times may be found deeper. Bubble populations under
breaking waves are very inhomogeneous in time and space. Bubbles can be observed to
split under several breakup mechanisms, including turbulent induced breakup, shear-driven
breakup resonant oscillation and tip-streaming [35, 57, 169]. The dynamics of a bubble
60 Environ Fluid Mech (2017) 17:47–77
123
under unsteady perturbations is affected by the balance between forces acting on the
bubble, but the residence time of the bubble in the turbulent flow field should also be taken
into account [31, 81]. A large amount of bubbles are generated and broken in the shear
zones, where velocity gradients play a great role in the breakup process. Current breakup
models are unable to quantify accurately the breakup rate and bubble size distributions in a
complete unsteady turbulent flow with high void fractions, partly because the two-phase
flow mixture beneath breaking waves is a highly unsteady process, never reaching an
equilibrium. As an illustration, Fig. 4 presents the normalized probability distribution
functions of bubble radii beneath a plunging seawater jet, at three vertical elevations. In
Fig. 4, each histogram column represents the probability of bubble radius in 0.25 mm
increments: that is, the probability of a bubble radius from 1.0 to 1.25 mm is represented
by the column labelled 1.0. The last column (i.e. 0.5) indicates the probability of bubble
radii large than 5 mm. The data showed a broad spectrum of bubble sizes at all elevations,
from less than 0.25 mm more than 5 mm (Fig. 4). The bubble radius distributions showed
a preponderance of small bubble sizes relative to the mean: the mode was between 0 and
0.5 mm, although the mean radius was about 1.5–2.3 mm (Table 1). If equilibrium exists,
the ‘equilibrium’ bubble size distribution is dictated by all the events taking place in the
flow, in particular by the relative rates of bubble breakup and bubble coalescence, the
bubble-turbulence interactions and the bubble clustering.
Table 1 Summary of bubble size observations
Flowtopology
Description Equivalent bubbleradius (mm)
Reference
Breakingwaves
Open ocean (Fig. 14) 0.8–1 Bowyer [20]
Open ocean (Fig. 6)Laboratory focused waves paddle (10 cmplunging) (Fig. 4)
1 Dean and Stokes[67]
Open ocean 0.7 Stokes et al. [166]
LaboratoryWind ? paddle (*12 cm breaking height)Fresh (but not clean) water
1–1.7 Leifer et al. [113]
Laboratory focused waves paddle (*20 cmplunging)
Fresh water
1.5 Rojas and Loewen[156]
Laboratory wave paddle sloping beachspilling—plunging (12.6–16.5 cm mid-scale/5.0–6.5 small-scale)
2.01 (mid-scale)–4.28(small-scale)
Mori et al. [135]
Laboratory wave paddle submerged reefspilling—plunging (9.5 cm)
Tap water
1.7–2.4 Blenkinsopp andChaplin [15]
Plungingjet
Circular plunging jet (Ø = 25, 12.5 and6.85 mm)
Tap water (Table 3)
4.7 (Ø = 25 mm)2.2–3.7(Ø = 12.5 mm)
1.7–2 (Ø = 6.85 mm)
Chanson et al. [32]
Circular plunging jet (Ø = 12.5 mm)Tap water, saltwater, seawater (Table 5)
2.2–3.1 (tap water)2.3–3.3 (saltwater)1.5–2.3 (seawater)
Chanson et al. [48]
Environ Fluid Mech (2017) 17:47–77 61
123
4.2 Water solution: freshwater, saltwater or seawater?
A number of studies tested the influence of salinity on the air bubble entrainment, although
there is no agreement between most published results. A majority of such studies tested
experimentally saltwater solutions, typically by gradually adding salt to freshwater: i.e.,
synthetic seawater [20, 28, 122, 160] (Also Table 1). Differences have been attributed to
some form of inhibited coalescence process in saltwater. Controlled experiments with
genuine seawater are rare (e.g. [16, 25, 48]). In strongly aerated regions (void fractions
[2 %), the results showed differences between freshwater, saltwater (synthetic seawater)
and genuine seawater in terms of void fraction and bubble sizes. Despite differences in
trends, an overall conclusion was that large amount of air bubbles were entrained in all
solutions, and the majority of bubbles in the aerated flow region had radii on the order of a
millimetre (Fig. 4).
It must be stressed however that the rare data were obtained with different geometric
scales. In air–water flows, experimental results demonstrated that scale effects may be
significant [31, 42, 45, 151, 184], in particular in terms of bubble numbers and sizes. In
turn, small scale experiments are more sensitive to changes in water salinity, and this may
bias the findings.
4.3 Froude number and definition
The definition of the Froude number is of paramount importance, because it is used to
detect the breaking event onset and termination during the evolution of the waves in the
surf zone. It is also the critical parameter allowing scaling of the data, assuming analogies
between hydraulic jumps, tidal bores and rollers due to breaking waves. Table 2 regroups a
number of definitions. Several criteria have been used in the past to detect the inception of
wave breaking. Wave steepness, based on Stokes’ limiting wave height, has been widely
used as an proxy for wave breaking. The occurrence of breaking has also been related to
the energy content in the waves, to the acceleration of the free-surface, or to the
Bubble radius (mm)
FDP
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.03
0.06
0.09
0.12
0.15
0.1810 mm - 3673 bubbles15 mm - 3080 bubbles25 mm - 1808 bubbles
Fig. 4 Pseudo-bubble radius distributions in seawater generated by a vertical circular plunging jet, at 10, 15and 25 mm beneath the free-surface—Data set: Chanson et al. [48], jet diameter: 12.5 mm, impact velocity:3.1 m/s, water solution: Pacific Ocean seawater off Japan coastline—All bubble sizes were recorded in air–water regions with void fractions greater than 2 %
62 Environ Fluid Mech (2017) 17:47–77
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prescription of a limiting value to the fluid velocity at the crest of the wave prior to
breaking (horizontal fluid velocity: [146] (see Fig. 2); [181]; vertical fluid velocity: [103]).
However consensus and agreement has not been reached for the establishment of a simple,
accurate and universal valid criterion. This is in part because wave breaking is a highly
complicated process involving highly non-linear effects including varying bathymetries,
currents, and wind (not taken into account in any study). An improvement in the quest has
been the use of a Relative Trough Froude Number (RTFN, [145]), based on the analogy
with a moving hydraulic jump in a one-dimensional open channel flow. The Froude
number can be obtained, as detailed in Table 2, by the mean of the P-theorem, momentum
considerations (Appendix 2) and by simple construction considering the quantities avail-
able. A discussion is provided in Appendix 2, since consistency and physical adequacy are
key features of this study. It is also important to consider the quantities available in the
various sources of data (Table 2, third column) (Fig. 5).
The strength of a breaking roller and the rate of energy dissipation in the roller are
directly linked to the Froude number [5, 21]. This is detailed in Appendix 2. The roller
length Lr increases monotonically with increasing Froude number, as illustrated in Fig. 6a.
For stationary hydraulic jumps, very detailed experimental measurements yielded:
Table 2 Froude number definitions in breaking waves, bores, surges and jumps
Froude numberdefinition
Reference Quantities Comment
Fr = (U ? V1)/(g 9 d1)
1/2Bazin [11],Boussinesq[19]
Implies the knowledge of thecurrent condition V1 and d1flowing adversely the roller,together with the celerity of theroller toe U
Classical definition, satisfying themass and momentumconservation equations, in arectangular horizontal channel
Fr = (U ? V1)/(g 9 A1/B1)
1/
2
Chanson [41] Implies further the knowledge ofthe initial cross-section area A1
and free-surface width B1
Application of mass andmomentum conservationequations for an irregular-shapedchannel
Fr = (ccrest -utrough)/ctrough
Okamoto andBasco [145]
Velocity of the crest, trough anddepth averaged velocity in thetrough region
Can be applied to deep waterbreaking waves
Fr = (((2 9 d2/d1 ? 1)2 -1)/8)1/2
Tissier et al.[171]
Water heights upstream anddownstream the jump
Not accurate enough [3]
Fr = (1 ? 3/2 9 Hr/d1 ? 1/2 9 (Hr/d1)
2)1/2
Tonelli andPetti [172]
Same as the previous definition,but written as a function of theroller height and water depth infront of the roller
Not accurate enough [3]
Fr = us/cb Bacigaluppiet al. [3]
Cheng andWang [53]
us is the free surface velocity at thecrest of the wave, and cb thewave celerity
Implies calibration. Fr = 0.75 to 1found to be the value whenwaves start breaking [3, 53].
Problem: based on the idea thatwave breaking occurs when thefree surface velocity exceeds thewave celerity (which is not truefor spilling, and true for thecelerity inside the crest forplunging)
Environ Fluid Mech (2017) 17:47–77 63
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Lr
d1¼ 6� Fr� 1ð Þ ð4Þ
where d1 is the upstream depth and Fr = (U ? V1)/(g 9 d1)1/2 (see symbol definition in
Fig. 1). Equation (4) is compared to stationary hydraulic jump and tidal bore data in
Fig. 6a. Practically another relevant quantity is the length of the aerated region Lair.
Typical experimental observations are reported in Fig. 6b, suggesting that the aeration
region length may be about 1.5–2 times longer than the roller itself.
Another measure for the roller strength is the height of the roller hr. Indeed the height of
the roller increases with increasing Froude number. Basic momentum considerations show
a unique relationship between roller height and Froude number for Fr[ 2 (Appendix 2):
hr
d1¼ 1
2�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ 8� Fr2p
� 3� �
Fr [ 2 ð5Þ
For Fr\ 2, physical observations in tidal bores showed that the surface immediately
upstream of the roller was curved upwards and the roller height was less than the difference
in conjugate depths (Fig. 7). This is illustrated in Fig. 7, by comparing experimental data
and Eq. (5). For breaking bore (i.e. 1.4\ Fr\ 2), the roller height data were best cor-
related by
hr
d1¼ 1:59� Fr� 1:16ð Þ0:94 1:4 \Fr \2 ð6Þ
The mean slope of the breaking roller can be obtained tan/ = hr/Lr. When the height of
the breaker hr is lower than the difference in conjugate depths (d2 - d1), the front face of
the breaking wave may not be fully saturated. Importantly the evolution of the roller is
highly unsteady and depends on the evolution of many parameters, including the variation
of the bathymetry or the current for example. Hence some simplistic parameters such as the
roller height might not characterise accurately the breaking wave roller.
Lr/hr
snoitavresb ofob
N
0 1 2 3 4 5 6 7 8 90
2
4
6
8
10
12
14
16Steady breakerTidal boreHydraulic jump
Fig. 5 Probability distributionsfunctions of roller aspect ratio Lr/hr—Re-analysis of large-scalewave breaking experiments [58,87], stationary hydraulic jumpdata [106, 138, 139, 178, 180],and tidal bore data [51, 138]
64 Environ Fluid Mech (2017) 17:47–77
123
Fr
L r/d
1
1 2 3 4 5 6 7 8 9 10 11 120
10
20
30
40
50
60
70
80
90
100
110
Tidal bore laboratoryTidal bore SéluneMurzyn et al (2007)Kucukali and Chanson (2008)Murzyn & Chanson (2009)Richard & Gavrilyuk (2013)Wang et al. (2015)Wang and Chanson (2015)6×(Fr-1)
Chanson (2010)Virey (2013)2.91×(d2/d1-1)
(B)
Fr
L air/
d 1
1 2 3 4 5 6 7 8 9 10 11 120
10
20
30
40
50
60
70
80
90
100
110
6×(Fr-1)Chanson (2010)Chachereau and Chanson (2011)Virey (2013)
(A)
Fig. 6 Roller length and aeration length in breaking hydraulic jumps and tidal bores as functions of theFroude number (Fr = (V1 ? U)/(g 9 d1)
1/2. a Roller length in breaking hydraulic jumps and tidal bores—Most data are hydraulic jump data unless indicated in the legend; tidal bore data by Chanson and Toi [51].b Aeration region length in hydraulic jumps
Environ Fluid Mech (2017) 17:47–77 65
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5 How can better flow analogies help?
This study clearly shows that more work needs to be done to elucidate the physics of the
unsteady motion of a breaking roller. The metrology is clearly an issue to get accurate and
reliable physical measurements in this rapidly-varied aerated turbulent flow. The scale
effects are also a point to be addressed, with a close look at the differences due to the water
salinity (tap water, saltwater, seawater). Some practice, such as the use of the Hinze’s [89]
scale, are also believed to be misleading, as the theory was originally developed under
assumptions not valid in the case of breaking waves. The analogy between hydraulic
jumps, breaking tidal bores and breaking waves may be more appropriate in order to
deduce quantities aimed at a modelling effort.
A stationary hydraulic jump is the sudden transition from a high-velocity free-surface flow
impinging into a slower open channel flow. The rapid transition, called the roller, is char-
acterised by spray and splashing with a highly fluctuating free-surface, together with highly-
aerated turbulent flow structures within, and a large amount of energy dissipation takes place
[5]. The application of the energy principle across the roller gives an expression of the energy
dissipation. For a smooth horizontal rectangular channel and assuming hydrostatic pressure
upstream and downstream of the roller, the rate of energy dissipation is [4, 21]:
DEE1
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ 8� Fr2p
� 3� �3
16�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ 8� Fr2p
� 1� �
� 1þ 12� Fr2
� �
ð7Þ
where E1 is the upstream total energy: E1 = d1 ? (V1)2/(2 9 g). The rate of energy dis-
sipation may exceed 70 % for Fr[ 9 [32, 88]. However a non-breaking undular jump is
observed at low Froude numbers. The transition from a stationary undular jump into a jump
with a marked breaking roller takes place with the progressive apparition of characteristic
three-dimensional flow features with increasing Froude numbers, including cross-waves
Fr
h r/d
1, (d
2-d 1
)/d1
1.2 1.4 1.6 1.8 2 2.20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
hr/d1 Leng & Chanson (2015)hr/d1 Koch & Chanson (2009)hr/d1 Chanson (2010)hr/d1 Docherty & Chanson (2012)(d2-d1)/d1=0.5×(sqrt(1+8×Fr2)-3)1.587×(Fr{-1.1612)0.9426
Fig. 7 Roller height in breakingtidal bores as function of theFroude number (Fr = (V1 ? U)/(g 9 d1)
1/2—Comparisonbetween breaking tidal bore dataand Eq. (5) (black solid line) andEq. (6) (red dashed line)
66 Environ Fluid Mech (2017) 17:47–77
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and cockscomb roller [46, 144]. A hydraulic jump with a fully-developed breaking roller is
observed only for Froude numbers greater than 2–4 [35, 158]. Thus a stationary jump
might not be truly representative of a spilling breaker, typically observed for Fr\ 2.
It is believed that a more relevant analogy is the tidal breaking bore, as seen in Fig. 2b.
A breaking bore with a quasi-two-dimensional roller is observed for Froude numbers
greater than 1.4–1.6 [104, 115], although localised form of breaking might appear for
Froude numbers above 1.3–1.4, including shock waves upstream of and limited breaking at
the first wave crest [37, 173]. In the field, tidal bores are typically observed for Froude
numbers less than 2.5, as shown in Table 3. As a further illustration, the Froude number
was estimated to be about 2 for the bore seen in Fig. 2b2. Based upon basic energy
considerations, the power dissipated in the bore roller may be estimated as (Appendix 2):
Pd ¼ q� g3=2 � d5=21 � L� Fr �
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ 8� Fr2p
� 3� �3
16�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ 8� Fr2p
� 1� � ð8Þ
where Pd is in Watts, L is the transverse length of the bore roller and where
E1 = d1 ? (V1 ? U)2/(2 9 g) for a tidal bore. For the tidal bore seen in Fig. 2b2, the power
dissipated was about: Pd &160 MW. When a tidal bore propagates upstream in a sloping
channel, experimental observations showed a complex transformation of a bore into a sta-
tionary hydraulic jump [30, 40]. In laboratory, the transformation could last from 5 to 10 min
up tomore than 20 min. At full scale, the duration of the processwould bemuch longer, based
upon a Froude similitude scaling, but the physical process remains poorly understood.
Even if the external forces generating the final roller are different in the flows considered in
this study, the general aspect of the turbulent aerated bores seem visually similar in terms of
lengths scales, intensities and aeration. Indeed, looking at the general hydrodynamic features of
the flows presented in the literature, the roller region results in a sequence of splash-ups,
inducing large-scale horizontal vortices. Considering the description of the splashing hydro-
dynamics in breaking waves (e.g. [18, 119, 132]), the roller region is roughly composed of 3–4
large horizontal vortices, which is consistent with experimental observations in tidal bores for
Fr\ 2. All the experimental studies, whatever the flow, indicated the capacity of successive
Table 3 Detailed field observations of tidal bores
Reference River/Estuary Date Bore type d1 (m) V1
(m/s)U(m/s)
Fr
Wolanski et al. [183] Daly 2/07/2003 Undular 1.50 0.15 4.7 1.04
Simpson et al. [163] Dee 6/09/2003 Breaking 0.72 0.15 4.1 1.79
Chanson et al. [50] Garonne 10/09/2010 Undular 1.77 0.33 4.5 1.30
11/09/2010 Undular 1.81 0.30 4.2 1.20
Mouaze et al. [138] Selune 24/09/2010 Breaking 0.38 0.86 2.0 2.35
25/09/2010 Breaking 0.33 0.59 2.0 2.48
Reungoat et al. [154] Garonne 7/06/2012 am Undular 2.72 0.68 3.8 1.02
7/06/2012 pm Undular 2.65 0.59 4.6 1.19
Furgerot et al. [80] See 7/05/2012 Undular 0.90 0.40 3.2 1.39
Reungoat et al. [155] Garonne 13/10/2013 Undular 2.05 0.26 4.3 1.27
Leng and Chanson [115] Qiantang 6/09/2013 Breaking 1 – 3.6 2.1
Chanson [43] Qiantang 12/10/2014 Breaking 2 to 2.5 – 4.5 2.00
Italic data rough estimate; (–) data not available
Environ Fluid Mech (2017) 17:47–77 67
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splash-ups to entrap air large quantities of air. The co-existence between the large-scale
clockwise vortices and some smaller counter-clockwise vortices, generated by backward
splashing and impingement occurring in between co-rotative vortices pairs, is reported in all the
observations. These large eddies are responsible for the turbulent fluctuations of the shapeof the
rollers.All the observations indicated that both air entrainment, turbulent shear and vorticity are
generated along the surface of the rollers and in the toe region. The total air volume is less
important than the number of bubbles and their actual size distributions in the roller region. The
study of the highly fluctuating dynamics of the roller region should then bring fruitful insight in
the dynamics of the flow. The temporal evolution of this region is highly required to get a better
description of the variation of the roller height and energy dissipation, as a function of time.
The quantification of some parameters may be improved and some physical description can
be highlighted, using the most appropriate analogy (roller/tidal bore) rather than the one typ-
ically used (roller/hydraulic jump). The roller length and height have been found to be fairly
similar between hydraulic jumps and tidal bores (Fig. 6), even comparing laboratory and field
studies, thus free of scale effects. Considering the tidal bore analogy and the corresponding
observations presented in Fig. 5, the length of the roller should be considered within 1.5\Lr/
hr\ 5.The transitionbetweenbreakingandnon-breakingwaveswould takeplace for Fr[ 1.3,
as observed in tidal bores. It was shown that the same limit value was inaccurately attributed to
the transition between undular and breaking hydraulic jumps, but most recent studies proved it
to be for Fr[ 2 in stationary jumps. It is also interesting to notice that the celerity of the bore
region considered in tidal bores is the celerity of the toe [114, 115], and not the celerity of the
crest as used in modelling studies. The celerity of a tidal bore is found to be different from the
celerity of a solitary wave. In a wave current problem, the Froude number represents the ratio
between the velocity of a wave travelling and that of the current. The definition of the Froude
number, as given for tidal bores in Table 2, should then be useful for breakingwaves studies. It
is similar to the estimation based on the RTFN concept [62, 145], and should be appropriate
since itwould be suitable for any type of analogy (deep and shallowwater breakingwaves). It is
noted that the celerity of the crest, considered byOkamoto and Basco [145] should be replaced
by the celerity of the toe, ctrough being V1 with the current notation.
This study also presents some acknowledged limitations. The current work cannot provide
any definition of a criterion for breaking onset. Experimental studies do not focus on the
initiation and evolution of the breaking region in tidal bores or hydraulic jumps. Estimations of
the wave velocity during breaking can lead to substantial discrepancies [146] and remains a
challenge. A steepening wave can propagate for a long distance before clearly breaking and
developing a roller; so the crest kinematics still need to be elucidated in complex situations
(3D, current, turbulence, wind, etc.). The transient evolution of the roller is crucial for future
studies [115]. Future interests should also encompass bubble size distribution and dynamics,
two-phase gas–liquid turbulence and vorticity generation in the roller region. Interestingly,
tidal bore is itself also a great example of highly non-linear phenomenon, as the free-surface
evolution of the undulations observed in rivers and laboratories do not show any accordance
with classical wave theories (cnoidal, solitary, Stokes). Analogies are complicated both ways.
6 Conclusions
In this study, we analysed and discussed the different analogies proposed in the literature tomodel
the breaking process of the waves in the surf zone. Numerical models not only need experimental
data for rigorousandextensivevalidations, but also fordevelopment.Even themost up-to-date and
68 Environ Fluid Mech (2017) 17:47–77
123
accurate numerical models are still limited by empirical aspects. So a better knowledge of the
temporal and spatial evolution of the aerated region under breaking waves is crucial.
Thequantitieswhichareneeded formost numericalmodels are acharacteristicFroudenumber
to detect the breaking process occurrence, the geometrical and dynamical characteristics of the
roller (height, length, mean front slope angle, celerity), the energy dissipation and the bubble size
distribution. Two features are reported to be of great importance in all experimental studies: the
bubble size distribution and the bubble cloud void fraction, the latter being highly dependent on
the accurate quantification of the number of bubbles. But this aspect of the flow remains a
technical challenge for both the numericalmodels and experimental technics. A number of issues
remain and definitive conclusions cannot be drawn. The actual best practices have been discussed
and better estimations have been proposed based on recent experimental data sets.We showed in
particular that measurements, performed in tidal bores will lead to improve the description of the
bubble plumes. The authors believe that the unsteady evolution of a decelerating tidal bore,
showing transition between a tidal bore to a stationary hydraulic jump [40], could bring intriguing
insights into the processes involved in the highly non-linear problem of breaking waves.
Acknowledgments The authors thank their students, former students and co-workers for their work andinput. The financial support of the Australian Research Council (Grant DP120100481) is acknowledged.
Appendix 1
See Table 4.
Table 4 Breaking waves, bores, surges and jumps: flow analogies in the literature
Analogy spilling breaker and steady breakers Banner and Phillips [6, 7], Duncan [73, 74],Banner and Peregrine [8]Cointe and Tulin [59]Lin and Rockwell [121]Dabiri and Gharib [63]
Analogy breaking wave and Plunging jet Cipriano and Blanchard [56]Hubbard et al. [93]Chanson and Cummings [44]Oguz et al. [143]Chanson et al. [47, 48]Salter et al. [160]
Analogy spilling breaker and hydraulic jump Longuet-Higgins [123]Peregrine and Svendsen [147]Madsen [128]Brocchini and Peregrine [23, 24]
Analogy spilling breaker and bore/positivesurge
Longuet-Higgins [123]Peregrine and Svendsen [147]Brocchini and Peregrine [24]
Analogy breaking bore/positive surge andhydraulic jumps
ALL Hydraulics Textbooks, dealing with open channelflows, as well as:
Darcy and Bazin [65]Stoker [164]Tricker [174]Henderson [88]Lighthill [118]Chanson [32, 41]
Other analogies—spilling breaker and densitycurrent
Longuet-Higgins and Turner [124]
Environ Fluid Mech (2017) 17:47–77 69
123
Appendix 2: Basic momentum and energy considerations for boresand jumps
A bore is an abrupt rise in water depth (Fig. 1) and its front may be analysed as a hydraulic
jump in translation [41, 118, 174]. In a system of reference in translation with the bore
front, the integral form of the continuity and momentum equations gives a relationship
between the cross-section area upstream and downstream of the roller, A1 and A2
respectively, and the upstream Froude number Fr1 [114]
Fr21 ¼1
2� A2
A1
� B1
B� 2� B0
B
� �
þ B0
B� A2
A1
� �
þ A2
A2 � A1
� Ffric �W � sinh
q� g� A21
B
ð9Þ
where Fr1 is the bore Froude number: Fr = (U ? V1)/(g 9 A1/B1)1/2, V is the cross-
sectional averaged velocity positive downstream as shown in Fig. 1, U is the bore celerity
positive upstream, the channel cross-sectional area A measured perpendicular to the flow
direction, q is the water density, g is the gravity acceleration, Ffric is the friction force, W is
the weight force, h the angle between the bed slope and horizontal, and the subscripts 1 and2 refer to the flow conditions immediately before and after the bore roller respectively
(Fig. 1). In Eq. (9), B and B0 are characteristic widths functions of the bathymetry:
B ¼ A2 � A1
d2 � d1ð10Þ
B0 ¼
R
A2
A1
R
q� g� d2 � zð Þ � dA
12� q� g� d2 � d1ð Þ2
ð11Þ
with d the flow depth (Fig. 1b). For a smooth rectangular horizontal channel, Eq. (9) yields
to the Belanger equation:
d2
d1¼ 1
2�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ 8� Fr21
q
� 1
� �
Smooth horizontal rectangular channel ð12Þ
where d1 and d2 are respectively the upstream and downstream flow depth, and Fr1 sim-
plifies into: Fr1 = (U ? V1)/(g 9 d1)1/2 (Fig. 1). First presented in 1841 [12, 36], Eq. (12)
is inappropriate in an irregular channel [41]. The roller height hr is in first approximation:
hr = d2 - d1 and Eq. (12) may rewritten as:
Fr21 ¼1
2� d2 � d1
d1
� �2
þ 3
2
d2 � d1
d1Smooth horizontal rectangular channel ð13Þ
Note that the above development (Eq. 9, 12 and 13) assumes implicitly that the bore
celerity is uniform and the roller shape two-dimensional. Field and laboratory observations
suggested that these assumptions are simplistic [114].
The application of the energy principle across the bore roller gives an expression of the
energy dissipation [4, 21]:
70 Environ Fluid Mech (2017) 17:47–77
123
DE ¼ d1 þV1 þ Uð Þ2
2� g� d2 +
V2 þ Uð Þ2
2� g
!
ð14Þ
assuming hydrostatic pressure upstream and downstream. For a smooth horizontal rect-
angular channel, the rate of energy dissipation becomes [5]:
DEE1
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ 8� Fr21
q
� 3
� �3
16�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ 8� Fr21
q
� 1
� �
� 1þ 12� Fr21
� �
ð15Þ
where E1 = d1 ? (V1 ? U)2/(2 9 g). The rate of energy dissipation ranges from 0 for
Fr1 = 1 to more than 70 % for Fr1[ 9 [32, 88]. The power dissipated in the bore is:
Pd ¼ q� g� V1 þ Uð Þ � d1 � L� DE ð16Þ
where L is the transverse length of the bore roller.In the above equations, the solution for a
stationary hydraulic jump is obtained for U = 0.
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